Maximum Satisfiability of Mission-Time Linear Temporal Logic

Gokul Hariharan, Phillip H. Jones, Kristin Yvonne Rozier, Tichakorn Wongpiromsarn

Abstract: Mission-time Linear Temporal Logic (MLTL) is a variant of Linear Temporal Logic (LTL) with finite interval bounds on temporal operators, and is a popular formal specification language for safety-critical cyber-physical systems. Given a set of specifications, the maximum satisfiability problem (MaxSAT) asks to find the maximum number of simultaneously satisfiable specifications. MaxSAT is useful for system designing complex systems, e.g., system design and feature prioritization. Considering the significant advances in MaxSAT for Boolean logic, we develop translations from MLTL to Boolean logic to solve the MLTL MaxSAT problem. Given an MLTL formula \(\varphi \) of length \(|\varphi |\) with maximum interval length m, our first (recursive) translator runs in \(\mathcal O(m^{|\varphi |})\) time. Our second, improved (iterative) translator runs in \(\mathcal O(|\varphi |^2m^2)\) time. Performance tests of satisfiability checks on MaxSAT instances illustrate that these Boolean translations perform significantly better than the best satisfiability checking approaches reported recently in the literature on real and random instances. Furthermore, the second translator is embarrassingly parallelizable to a factor of \(|\varphi |m\). We contribute to (1) an easy-to-implement translation from MLTL to Boolean logic that runs in \(\mathcal O(m^{|\varphi |})\) time, and (2) an efficient translation that runs in \(\mathcal O(|\varphi |^2m^2)\) time, and prove their correctness and runtime. Lastly, (3) we consider examples of using Boolean MaxSAT solvers to solve the MLTL MaxSAT problem.